I have a question which arises from looking at the impact free Boltzmann equation.

Let $(\vec{x},\vec{v})$ be a vector in our phase space $\Gamma^N = \mathbb{R}^{6N}$. The dynamics of a state are determined by the distribution function $f(\vec{x}, \vec{v}, t)$. Where $f(\vec{x}, \vec{v}, t) d^3x d^3v$ is the amount of particles at time $t$ in the volume element $d^3x d^3v$.

To derive the impact free Boltzmannequation we simply have to equate the time derivative of the volume-integral of $f$ to the flow of particles out of that volume (The amount of particles going out of a certain phasespace volume determine how the state goes on in time).

This is where my question arises. Why is the right side of the equation the flow of particles out of $dV$? $(\vec{v},\vec{a})$ is the time derivative of $(\vec{x}, \vec{v})$, but I still dont see it. Can somebody give me some pointers what I have to read about to get an intuitive and mathematical feeling for why this is right?